% TIMEFORMAT='%3R'; { time (exec 2>&1; /home/martin/bin/satallax -E /home/martin/.isabelle/contrib/e-2.5-1/x86_64-linux/eprover -p tstp -t 5 /home/martin/judgement-day/tptp-thf/tptp/FFT/prob_306__3225672_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:10:21.680

% Could-be-implicit typings (4)
thf(ty_n_t__Set__Oset_It__Nat__Onat_J, type,
    set_nat : $tType).
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Num__Onum, type,
    num : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (30)
thf(sy_c_FFT__Mirabelle__ulikgskiun_OIDFT, type,
    fFT_Mirabelle_IDFT : nat > (nat > complex) > nat > complex).
thf(sy_c_FFT__Mirabelle__ulikgskiun_Oroot, type,
    fFT_Mirabelle_root : nat > complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex, type,
    plus_plus_complex : complex > complex > complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum, type,
    plus_plus_num : num > num > num).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex, type,
    times_times_complex : complex > complex > complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat, type,
    times_times_nat : nat > nat > nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum, type,
    times_times_num : num > num > num).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex, type,
    uminus1204672759omplex : complex > complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Complex__Ocomplex, type,
    groups59700922omplex : (nat > complex) > set_nat > complex).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat, type,
    groups1842438620at_nat : (nat > nat) > set_nat > nat).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Num_Onum_OBit0, type,
    bit0 : num > num).
thf(sy_c_Num_Onum_OOne, type,
    one : num).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex, type,
    numera632737353omplex : num > complex).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat, type,
    numeral_numeral_nat : num > nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum, type,
    ord_less_num : num > num > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum, type,
    ord_less_eq_num : num > num > $o).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex, type,
    power_power_complex : complex > nat > complex).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat, type,
    power_power_nat : nat > nat > nat).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex, type,
    divide1210191872omplex : complex > complex > complex).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat, type,
    divide_divide_nat : nat > nat > nat).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat, type,
    set_or562006527an_nat : nat > nat > set_nat).
thf(sy_v_a, type,
    a : nat > complex).
thf(sy_v_i, type,
    i : nat).
thf(sy_v_m, type,
    m : nat).

% Relevant facts (119)
thf(fact_0_mbound, axiom,
    ((ord_less_nat @ zero_zero_nat @ m))). % mbound
thf(fact_1_ibound, axiom,
    ((ord_less_eq_nat @ m @ i))). % ibound
thf(fact_2_root__cancel1, axiom,
    ((![M : nat, I : nat, J : nat]: ((power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M)) @ (times_times_nat @ I @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J))) = (power_power_complex @ (fFT_Mirabelle_root @ M) @ (times_times_nat @ I @ J)))))). % root_cancel1
thf(fact_3_Power_Oring__1__class_Opower__minus__even, axiom,
    ((![A : complex, N : nat]: ((power_power_complex @ (uminus1204672759omplex @ A) @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = (power_power_complex @ A @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)))))). % Power.ring_1_class.power_minus_even
thf(fact_4_IDFT__def, axiom,
    ((fFT_Mirabelle_IDFT = (^[N2 : nat]: (^[A2 : nat > complex]: (^[I2 : nat]: (groups59700922omplex @ (^[K : nat]: (divide1210191872omplex @ (A2 @ K) @ (power_power_complex @ (fFT_Mirabelle_root @ N2) @ (times_times_nat @ I2 @ K)))) @ (set_or562006527an_nat @ zero_zero_nat @ N2)))))))). % IDFT_def
thf(fact_5_add__2__eq__Suc, axiom,
    ((![N : nat]: ((plus_plus_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N) = (suc @ (suc @ N)))))). % add_2_eq_Suc
thf(fact_6_add__2__eq__Suc_H, axiom,
    ((![N : nat]: ((plus_plus_nat @ N @ (numeral_numeral_nat @ (bit0 @ one))) = (suc @ (suc @ N)))))). % add_2_eq_Suc'
thf(fact_7_power2__minus, axiom,
    ((![A : complex]: ((power_power_complex @ (uminus1204672759omplex @ A) @ (numeral_numeral_nat @ (bit0 @ one))) = (power_power_complex @ A @ (numeral_numeral_nat @ (bit0 @ one))))))). % power2_minus
thf(fact_8_zero__eq__power2, axiom,
    ((![A : complex]: (((power_power_complex @ A @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_complex) = (A = zero_zero_complex))))). % zero_eq_power2
thf(fact_9_zero__eq__power2, axiom,
    ((![A : nat]: (((power_power_nat @ A @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_nat) = (A = zero_zero_nat))))). % zero_eq_power2
thf(fact_10_divide__eq__eq__numeral1_I2_J, axiom,
    ((![B : complex, W : num, A : complex]: (((divide1210191872omplex @ B @ (uminus1204672759omplex @ (numera632737353omplex @ W))) = A) = (((((~ (((uminus1204672759omplex @ (numera632737353omplex @ W)) = zero_zero_complex)))) => ((B = (times_times_complex @ A @ (uminus1204672759omplex @ (numera632737353omplex @ W))))))) & (((((uminus1204672759omplex @ (numera632737353omplex @ W)) = zero_zero_complex)) => ((A = zero_zero_complex))))))))). % divide_eq_eq_numeral1(2)
thf(fact_11_eq__divide__eq__numeral1_I2_J, axiom,
    ((![A : complex, B : complex, W : num]: ((A = (divide1210191872omplex @ B @ (uminus1204672759omplex @ (numera632737353omplex @ W)))) = (((((~ (((uminus1204672759omplex @ (numera632737353omplex @ W)) = zero_zero_complex)))) => (((times_times_complex @ A @ (uminus1204672759omplex @ (numera632737353omplex @ W))) = B)))) & (((((uminus1204672759omplex @ (numera632737353omplex @ W)) = zero_zero_complex)) => ((A = zero_zero_complex))))))))). % eq_divide_eq_numeral1(2)
thf(fact_12_div__mult__self1, axiom,
    ((![B : nat, A : nat, C : nat]: ((~ ((B = zero_zero_nat))) => ((divide_divide_nat @ (plus_plus_nat @ A @ (times_times_nat @ C @ B)) @ B) = (plus_plus_nat @ C @ (divide_divide_nat @ A @ B))))))). % div_mult_self1
thf(fact_13_div__mult__self2, axiom,
    ((![B : nat, A : nat, C : nat]: ((~ ((B = zero_zero_nat))) => ((divide_divide_nat @ (plus_plus_nat @ A @ (times_times_nat @ B @ C)) @ B) = (plus_plus_nat @ C @ (divide_divide_nat @ A @ B))))))). % div_mult_self2
thf(fact_14_div__mult__self3, axiom,
    ((![B : nat, C : nat, A : nat]: ((~ ((B = zero_zero_nat))) => ((divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ C @ B) @ A) @ B) = (plus_plus_nat @ C @ (divide_divide_nat @ A @ B))))))). % div_mult_self3
thf(fact_15_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_nat @ M) = (numeral_numeral_nat @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_16_numeral__le__iff, axiom,
    ((![M : num, N : num]: ((ord_less_eq_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (ord_less_eq_num @ M @ N))))). % numeral_le_iff
thf(fact_17_numeral__less__iff, axiom,
    ((![M : num, N : num]: ((ord_less_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (ord_less_num @ M @ N))))). % numeral_less_iff
thf(fact_18_numeral__times__numeral, axiom,
    ((![M : num, N : num]: ((times_times_complex @ (numera632737353omplex @ M) @ (numera632737353omplex @ N)) = (numera632737353omplex @ (times_times_num @ M @ N)))))). % numeral_times_numeral
thf(fact_19_numeral__times__numeral, axiom,
    ((![M : num, N : num]: ((times_times_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ (times_times_num @ M @ N)))))). % numeral_times_numeral
thf(fact_20_mult__numeral__left__semiring__numeral, axiom,
    ((![V : num, W : num, Z : complex]: ((times_times_complex @ (numera632737353omplex @ V) @ (times_times_complex @ (numera632737353omplex @ W) @ Z)) = (times_times_complex @ (numera632737353omplex @ (times_times_num @ V @ W)) @ Z))))). % mult_numeral_left_semiring_numeral
thf(fact_21_mult__numeral__left__semiring__numeral, axiom,
    ((![V : num, W : num, Z : nat]: ((times_times_nat @ (numeral_numeral_nat @ V) @ (times_times_nat @ (numeral_numeral_nat @ W) @ Z)) = (times_times_nat @ (numeral_numeral_nat @ (times_times_num @ V @ W)) @ Z))))). % mult_numeral_left_semiring_numeral
thf(fact_22_numeral__plus__numeral, axiom,
    ((![M : num, N : num]: ((plus_plus_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ (plus_plus_num @ M @ N)))))). % numeral_plus_numeral
thf(fact_23_add__numeral__left, axiom,
    ((![V : num, W : num, Z : nat]: ((plus_plus_nat @ (numeral_numeral_nat @ V) @ (plus_plus_nat @ (numeral_numeral_nat @ W) @ Z)) = (plus_plus_nat @ (numeral_numeral_nat @ (plus_plus_num @ V @ W)) @ Z))))). % add_numeral_left
thf(fact_24_neg__numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((uminus1204672759omplex @ (numera632737353omplex @ M)) = (uminus1204672759omplex @ (numera632737353omplex @ N))) = (M = N))))). % neg_numeral_eq_iff
thf(fact_25_num__double, axiom,
    ((![N : num]: ((times_times_num @ (bit0 @ one) @ N) = (bit0 @ N))))). % num_double
thf(fact_26_nat__power__eq__Suc__0__iff, axiom,
    ((![X : nat, M : nat]: (((power_power_nat @ X @ M) = (suc @ zero_zero_nat)) = (((M = zero_zero_nat)) | ((X = (suc @ zero_zero_nat)))))))). % nat_power_eq_Suc_0_iff
thf(fact_27_power__Suc__0, axiom,
    ((![N : nat]: ((power_power_nat @ (suc @ zero_zero_nat) @ N) = (suc @ zero_zero_nat))))). % power_Suc_0
thf(fact_28_div__by__Suc__0, axiom,
    ((![M : nat]: ((divide_divide_nat @ M @ (suc @ zero_zero_nat)) = M)))). % div_by_Suc_0
thf(fact_29_power__mult__numeral, axiom,
    ((![A : complex, M : num, N : num]: ((power_power_complex @ (power_power_complex @ A @ (numeral_numeral_nat @ M)) @ (numeral_numeral_nat @ N)) = (power_power_complex @ A @ (numeral_numeral_nat @ (times_times_num @ M @ N))))))). % power_mult_numeral
thf(fact_30_power__mult__numeral, axiom,
    ((![A : nat, M : num, N : num]: ((power_power_nat @ (power_power_nat @ A @ (numeral_numeral_nat @ M)) @ (numeral_numeral_nat @ N)) = (power_power_nat @ A @ (numeral_numeral_nat @ (times_times_num @ M @ N))))))). % power_mult_numeral
thf(fact_31_div__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => ((divide_divide_nat @ M @ N) = zero_zero_nat))))). % div_less
thf(fact_32_nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ X @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % nat_zero_less_power_iff
thf(fact_33_div__mult__mult1__if, axiom,
    ((![C : nat, A : nat, B : nat]: (((C = zero_zero_nat) => ((divide_divide_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B)) = zero_zero_nat)) & ((~ ((C = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B)) = (divide_divide_nat @ A @ B))))))). % div_mult_mult1_if
thf(fact_34_div__mult__mult2, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C)) = (divide_divide_nat @ A @ B)))))). % div_mult_mult2
thf(fact_35_div__mult__mult1, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B)) = (divide_divide_nat @ A @ B)))))). % div_mult_mult1
thf(fact_36_distrib__right__numeral, axiom,
    ((![A : complex, B : complex, V : num]: ((times_times_complex @ (plus_plus_complex @ A @ B) @ (numera632737353omplex @ V)) = (plus_plus_complex @ (times_times_complex @ A @ (numera632737353omplex @ V)) @ (times_times_complex @ B @ (numera632737353omplex @ V))))))). % distrib_right_numeral
thf(fact_37_distrib__right__numeral, axiom,
    ((![A : nat, B : nat, V : num]: ((times_times_nat @ (plus_plus_nat @ A @ B) @ (numeral_numeral_nat @ V)) = (plus_plus_nat @ (times_times_nat @ A @ (numeral_numeral_nat @ V)) @ (times_times_nat @ B @ (numeral_numeral_nat @ V))))))). % distrib_right_numeral
thf(fact_38_distrib__left__numeral, axiom,
    ((![V : num, B : complex, C : complex]: ((times_times_complex @ (numera632737353omplex @ V) @ (plus_plus_complex @ B @ C)) = (plus_plus_complex @ (times_times_complex @ (numera632737353omplex @ V) @ B) @ (times_times_complex @ (numera632737353omplex @ V) @ C)))))). % distrib_left_numeral
thf(fact_39_distrib__left__numeral, axiom,
    ((![V : num, B : nat, C : nat]: ((times_times_nat @ (numeral_numeral_nat @ V) @ (plus_plus_nat @ B @ C)) = (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ V) @ B) @ (times_times_nat @ (numeral_numeral_nat @ V) @ C)))))). % distrib_left_numeral
thf(fact_40_mult__neg__numeral__simps_I1_J, axiom,
    ((![M : num, N : num]: ((times_times_complex @ (uminus1204672759omplex @ (numera632737353omplex @ M)) @ (uminus1204672759omplex @ (numera632737353omplex @ N))) = (numera632737353omplex @ (times_times_num @ M @ N)))))). % mult_neg_numeral_simps(1)
thf(fact_41_mult__neg__numeral__simps_I2_J, axiom,
    ((![M : num, N : num]: ((times_times_complex @ (uminus1204672759omplex @ (numera632737353omplex @ M)) @ (numera632737353omplex @ N)) = (uminus1204672759omplex @ (numera632737353omplex @ (times_times_num @ M @ N))))))). % mult_neg_numeral_simps(2)
thf(fact_42_mult__neg__numeral__simps_I3_J, axiom,
    ((![M : num, N : num]: ((times_times_complex @ (numera632737353omplex @ M) @ (uminus1204672759omplex @ (numera632737353omplex @ N))) = (uminus1204672759omplex @ (numera632737353omplex @ (times_times_num @ M @ N))))))). % mult_neg_numeral_simps(3)
thf(fact_43_add__neg__numeral__simps_I3_J, axiom,
    ((![M : num, N : num]: ((plus_plus_complex @ (uminus1204672759omplex @ (numera632737353omplex @ M)) @ (uminus1204672759omplex @ (numera632737353omplex @ N))) = (uminus1204672759omplex @ (plus_plus_complex @ (numera632737353omplex @ M) @ (numera632737353omplex @ N))))))). % add_neg_numeral_simps(3)
thf(fact_44_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_complex @ zero_zero_complex @ (suc @ N)) = zero_zero_complex)))). % power_0_Suc
thf(fact_45_power__0__Suc, axiom,
    ((![N : nat]: ((power_power_nat @ zero_zero_nat @ (suc @ N)) = zero_zero_nat)))). % power_0_Suc
thf(fact_46_power__zero__numeral, axiom,
    ((![K2 : num]: ((power_power_complex @ zero_zero_complex @ (numeral_numeral_nat @ K2)) = zero_zero_complex)))). % power_zero_numeral
thf(fact_47_power__zero__numeral, axiom,
    ((![K2 : num]: ((power_power_nat @ zero_zero_nat @ (numeral_numeral_nat @ K2)) = zero_zero_nat)))). % power_zero_numeral
thf(fact_48_power__add__numeral2, axiom,
    ((![A : complex, M : num, N : num, B : complex]: ((times_times_complex @ (power_power_complex @ A @ (numeral_numeral_nat @ M)) @ (times_times_complex @ (power_power_complex @ A @ (numeral_numeral_nat @ N)) @ B)) = (times_times_complex @ (power_power_complex @ A @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))) @ B))))). % power_add_numeral2
thf(fact_49_power__add__numeral2, axiom,
    ((![A : nat, M : num, N : num, B : nat]: ((times_times_nat @ (power_power_nat @ A @ (numeral_numeral_nat @ M)) @ (times_times_nat @ (power_power_nat @ A @ (numeral_numeral_nat @ N)) @ B)) = (times_times_nat @ (power_power_nat @ A @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))) @ B))))). % power_add_numeral2
thf(fact_50_power__add__numeral, axiom,
    ((![A : complex, M : num, N : num]: ((times_times_complex @ (power_power_complex @ A @ (numeral_numeral_nat @ M)) @ (power_power_complex @ A @ (numeral_numeral_nat @ N))) = (power_power_complex @ A @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))))))). % power_add_numeral
thf(fact_51_power__add__numeral, axiom,
    ((![A : nat, M : num, N : num]: ((times_times_nat @ (power_power_nat @ A @ (numeral_numeral_nat @ M)) @ (power_power_nat @ A @ (numeral_numeral_nat @ N))) = (power_power_nat @ A @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))))))). % power_add_numeral
thf(fact_52_power__Suc0__right, axiom,
    ((![A : complex]: ((power_power_complex @ A @ (suc @ zero_zero_nat)) = A)))). % power_Suc0_right
thf(fact_53_power__Suc0__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ (suc @ zero_zero_nat)) = A)))). % power_Suc0_right
thf(fact_54_Suc__numeral, axiom,
    ((![N : num]: ((suc @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ (plus_plus_num @ N @ one)))))). % Suc_numeral
thf(fact_55_div__mult__self1__is__m, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((divide_divide_nat @ (times_times_nat @ N @ M) @ N) = M))))). % div_mult_self1_is_m
thf(fact_56_div__mult__self__is__m, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((divide_divide_nat @ (times_times_nat @ M @ N) @ N) = M))))). % div_mult_self_is_m
thf(fact_57_eq__divide__eq__numeral1_I1_J, axiom,
    ((![A : complex, B : complex, W : num]: ((A = (divide1210191872omplex @ B @ (numera632737353omplex @ W))) = (((((~ (((numera632737353omplex @ W) = zero_zero_complex)))) => (((times_times_complex @ A @ (numera632737353omplex @ W)) = B)))) & (((((numera632737353omplex @ W) = zero_zero_complex)) => ((A = zero_zero_complex))))))))). % eq_divide_eq_numeral1(1)
thf(fact_58_divide__eq__eq__numeral1_I1_J, axiom,
    ((![B : complex, W : num, A : complex]: (((divide1210191872omplex @ B @ (numera632737353omplex @ W)) = A) = (((((~ (((numera632737353omplex @ W) = zero_zero_complex)))) => ((B = (times_times_complex @ A @ (numera632737353omplex @ W)))))) & (((((numera632737353omplex @ W) = zero_zero_complex)) => ((A = zero_zero_complex))))))))). % divide_eq_eq_numeral1(1)
thf(fact_59_div__mult__self4, axiom,
    ((![B : nat, C : nat, A : nat]: ((~ ((B = zero_zero_nat))) => ((divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ B @ C) @ A) @ B) = (plus_plus_nat @ C @ (divide_divide_nat @ A @ B))))))). % div_mult_self4
thf(fact_60_power__eq__0__iff, axiom,
    ((![A : complex, N : nat]: (((power_power_complex @ A @ N) = zero_zero_complex) = (((A = zero_zero_complex)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_61_power__eq__0__iff, axiom,
    ((![A : nat, N : nat]: (((power_power_nat @ A @ N) = zero_zero_nat) = (((A = zero_zero_nat)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_62_div2__Suc__Suc, axiom,
    ((![M : nat]: ((divide_divide_nat @ (suc @ (suc @ M)) @ (numeral_numeral_nat @ (bit0 @ one))) = (suc @ (divide_divide_nat @ M @ (numeral_numeral_nat @ (bit0 @ one)))))))). % div2_Suc_Suc
thf(fact_63_add__self__div__2, axiom,
    ((![M : nat]: ((divide_divide_nat @ (plus_plus_nat @ M @ M) @ (numeral_numeral_nat @ (bit0 @ one))) = M)))). % add_self_div_2
thf(fact_64_power__mono__iff, axiom,
    ((![A : nat, B : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)) = (ord_less_eq_nat @ A @ B)))))))). % power_mono_iff
thf(fact_65_power2__eq__iff__nonneg, axiom,
    ((![X : nat, Y : nat]: ((ord_less_eq_nat @ zero_zero_nat @ X) => ((ord_less_eq_nat @ zero_zero_nat @ Y) => (((power_power_nat @ X @ (numeral_numeral_nat @ (bit0 @ one))) = (power_power_nat @ Y @ (numeral_numeral_nat @ (bit0 @ one)))) = (X = Y))))))). % power2_eq_iff_nonneg
thf(fact_66_nat__power__less__imp__less, axiom,
    ((![I : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ I) => ((ord_less_nat @ (power_power_nat @ I @ M) @ (power_power_nat @ I @ N)) => (ord_less_nat @ M @ N)))))). % nat_power_less_imp_less
thf(fact_67_div__le__mono, axiom,
    ((![M : nat, N : nat, K2 : nat]: ((ord_less_eq_nat @ M @ N) => (ord_less_eq_nat @ (divide_divide_nat @ M @ K2) @ (divide_divide_nat @ N @ K2)))))). % div_le_mono
thf(fact_68_Euclidean__Division_Odiv__eq__0__iff, axiom,
    ((![M : nat, N : nat]: (((divide_divide_nat @ M @ N) = zero_zero_nat) = (((ord_less_nat @ M @ N)) | ((N = zero_zero_nat))))))). % Euclidean_Division.div_eq_0_iff
thf(fact_69_div__le__mono2, axiom,
    ((![M : nat, N : nat, K2 : nat]: ((ord_less_nat @ zero_zero_nat @ M) => ((ord_less_eq_nat @ M @ N) => (ord_less_eq_nat @ (divide_divide_nat @ K2 @ N) @ (divide_divide_nat @ K2 @ M))))))). % div_le_mono2
thf(fact_70_Suc__div__le__mono, axiom,
    ((![M : nat, N : nat]: (ord_less_eq_nat @ (divide_divide_nat @ M @ N) @ (divide_divide_nat @ (suc @ M) @ N))))). % Suc_div_le_mono
thf(fact_71_div__le__dividend, axiom,
    ((![M : nat, N : nat]: (ord_less_eq_nat @ (divide_divide_nat @ M @ N) @ M)))). % div_le_dividend
thf(fact_72_div__greater__zero__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (divide_divide_nat @ M @ N)) = (((ord_less_eq_nat @ N @ M)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % div_greater_zero_iff
thf(fact_73_less__mult__imp__div__less, axiom,
    ((![M : nat, I : nat, N : nat]: ((ord_less_nat @ M @ (times_times_nat @ I @ N)) => (ord_less_nat @ (divide_divide_nat @ M @ N) @ I))))). % less_mult_imp_div_less
thf(fact_74_div__times__less__eq__dividend, axiom,
    ((![M : nat, N : nat]: (ord_less_eq_nat @ (times_times_nat @ (divide_divide_nat @ M @ N) @ N) @ M)))). % div_times_less_eq_dividend
thf(fact_75_times__div__less__eq__dividend, axiom,
    ((![N : nat, M : nat]: (ord_less_eq_nat @ (times_times_nat @ N @ (divide_divide_nat @ M @ N)) @ M)))). % times_div_less_eq_dividend
thf(fact_76_div__nat__eqI, axiom,
    ((![N : nat, Q : nat, M : nat]: ((ord_less_eq_nat @ (times_times_nat @ N @ Q) @ M) => ((ord_less_nat @ M @ (times_times_nat @ N @ (suc @ Q))) => ((divide_divide_nat @ M @ N) = Q)))))). % div_nat_eqI
thf(fact_77_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_78_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat))). % le_numeral_extra(3)
thf(fact_79_add__One__commute, axiom,
    ((![N : num]: ((plus_plus_num @ one @ N) = (plus_plus_num @ N @ one))))). % add_One_commute
thf(fact_80_div__mult2__eq, axiom,
    ((![M : nat, N : nat, Q : nat]: ((divide_divide_nat @ M @ (times_times_nat @ N @ Q)) = (divide_divide_nat @ (divide_divide_nat @ M @ N) @ Q))))). % div_mult2_eq
thf(fact_81_power__less__imp__less__base, axiom,
    ((![A : nat, N : nat, B : nat]: ((ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (ord_less_nat @ A @ B)))))). % power_less_imp_less_base
thf(fact_82_power__gt__expt, axiom,
    ((![N : nat, K2 : nat]: ((ord_less_nat @ (suc @ zero_zero_nat) @ N) => (ord_less_nat @ K2 @ (power_power_nat @ N @ K2)))))). % power_gt_expt
thf(fact_83_nat__one__le__power, axiom,
    ((![I : nat, N : nat]: ((ord_less_eq_nat @ (suc @ zero_zero_nat) @ I) => (ord_less_eq_nat @ (suc @ zero_zero_nat) @ (power_power_nat @ I @ N)))))). % nat_one_le_power
thf(fact_84_power__strict__mono, axiom,
    ((![A : nat, B : nat, N : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)))))))). % power_strict_mono
thf(fact_85_split__div_H, axiom,
    ((![P : nat > $o, M : nat, N : nat]: ((P @ (divide_divide_nat @ M @ N)) = (((((N = zero_zero_nat)) & ((P @ zero_zero_nat)))) | ((?[Q2 : nat]: (((((ord_less_eq_nat @ (times_times_nat @ N @ Q2) @ M)) & ((ord_less_nat @ M @ (times_times_nat @ N @ (suc @ Q2)))))) & ((P @ Q2)))))))))). % split_div'
thf(fact_86_power__eq__imp__eq__base, axiom,
    ((![A : nat, N : nat, B : nat]: (((power_power_nat @ A @ N) = (power_power_nat @ B @ N)) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => ((ord_less_nat @ zero_zero_nat @ N) => (A = B)))))))). % power_eq_imp_eq_base
thf(fact_87_power__eq__iff__eq__base, axiom,
    ((![N : nat, A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (((power_power_nat @ A @ N) = (power_power_nat @ B @ N)) = (A = B)))))))). % power_eq_iff_eq_base
thf(fact_88_power2__nat__le__imp__le, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ (power_power_nat @ M @ (numeral_numeral_nat @ (bit0 @ one))) @ N) => (ord_less_eq_nat @ M @ N))))). % power2_nat_le_imp_le
thf(fact_89_power2__nat__le__eq__le, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ (power_power_nat @ M @ (numeral_numeral_nat @ (bit0 @ one))) @ (power_power_nat @ N @ (numeral_numeral_nat @ (bit0 @ one)))) = (ord_less_eq_nat @ M @ N))))). % power2_nat_le_eq_le
thf(fact_90_self__le__ge2__pow, axiom,
    ((![K2 : nat, M : nat]: ((ord_less_eq_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ K2) => (ord_less_eq_nat @ M @ (power_power_nat @ K2 @ M)))))). % self_le_ge2_pow
thf(fact_91_dividend__less__times__div, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ M @ (plus_plus_nat @ N @ (times_times_nat @ N @ (divide_divide_nat @ M @ N)))))))). % dividend_less_times_div
thf(fact_92_dividend__less__div__times, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ M @ (plus_plus_nat @ N @ (times_times_nat @ (divide_divide_nat @ M @ N) @ N))))))). % dividend_less_div_times
thf(fact_93_split__div, axiom,
    ((![P : nat > $o, M : nat, N : nat]: ((P @ (divide_divide_nat @ M @ N)) = (((((N = zero_zero_nat)) => ((P @ zero_zero_nat)))) & ((((~ ((N = zero_zero_nat)))) => ((![I2 : nat]: (![J2 : nat]: (((ord_less_nat @ J2 @ N)) => ((((M = (plus_plus_nat @ (times_times_nat @ N @ I2) @ J2))) => ((P @ I2))))))))))))))). % split_div
thf(fact_94_not__numeral__less__zero, axiom,
    ((![N : num]: (~ ((ord_less_nat @ (numeral_numeral_nat @ N) @ zero_zero_nat)))))). % not_numeral_less_zero
thf(fact_95_zero__less__numeral, axiom,
    ((![N : num]: (ord_less_nat @ zero_zero_nat @ (numeral_numeral_nat @ N))))). % zero_less_numeral
thf(fact_96_not__numeral__le__zero, axiom,
    ((![N : num]: (~ ((ord_less_eq_nat @ (numeral_numeral_nat @ N) @ zero_zero_nat)))))). % not_numeral_le_zero
thf(fact_97_zero__le__numeral, axiom,
    ((![N : num]: (ord_less_eq_nat @ zero_zero_nat @ (numeral_numeral_nat @ N))))). % zero_le_numeral
thf(fact_98_zero__less__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ A) => (ord_less_nat @ zero_zero_nat @ (power_power_nat @ A @ N)))))). % zero_less_power
thf(fact_99_zero__le__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => (ord_less_eq_nat @ zero_zero_nat @ (power_power_nat @ A @ N)))))). % zero_le_power
thf(fact_100_power__mono, axiom,
    ((![A : nat, B : nat, N : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ zero_zero_nat @ A) => (ord_less_eq_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N))))))). % power_mono
thf(fact_101_div__mult2__numeral__eq, axiom,
    ((![A : nat, K2 : num, L : num]: ((divide_divide_nat @ (divide_divide_nat @ A @ (numeral_numeral_nat @ K2)) @ (numeral_numeral_nat @ L)) = (divide_divide_nat @ A @ (numeral_numeral_nat @ (times_times_num @ K2 @ L))))))). % div_mult2_numeral_eq
thf(fact_102_sum__add__split__nat__ivl, axiom,
    ((![M : nat, K2 : nat, N : nat, G : nat > nat, F : nat > nat, H : nat > nat]: ((ord_less_eq_nat @ M @ K2) => ((ord_less_eq_nat @ K2 @ N) => ((![I3 : nat]: ((ord_less_eq_nat @ M @ I3) => ((ord_less_nat @ I3 @ K2) => ((G @ I3) = (F @ I3))))) => ((![I3 : nat]: ((ord_less_eq_nat @ K2 @ I3) => ((ord_less_nat @ I3 @ N) => ((H @ I3) = (F @ I3))))) => ((plus_plus_nat @ (groups1842438620at_nat @ G @ (set_or562006527an_nat @ M @ K2)) @ (groups1842438620at_nat @ H @ (set_or562006527an_nat @ K2 @ N))) = (groups1842438620at_nat @ F @ (set_or562006527an_nat @ M @ N)))))))))). % sum_add_split_nat_ivl
thf(fact_103_sum__add__split__nat__ivl, axiom,
    ((![M : nat, K2 : nat, N : nat, G : nat > complex, F : nat > complex, H : nat > complex]: ((ord_less_eq_nat @ M @ K2) => ((ord_less_eq_nat @ K2 @ N) => ((![I3 : nat]: ((ord_less_eq_nat @ M @ I3) => ((ord_less_nat @ I3 @ K2) => ((G @ I3) = (F @ I3))))) => ((![I3 : nat]: ((ord_less_eq_nat @ K2 @ I3) => ((ord_less_nat @ I3 @ N) => ((H @ I3) = (F @ I3))))) => ((plus_plus_complex @ (groups59700922omplex @ G @ (set_or562006527an_nat @ M @ K2)) @ (groups59700922omplex @ H @ (set_or562006527an_nat @ K2 @ N))) = (groups59700922omplex @ F @ (set_or562006527an_nat @ M @ N)))))))))). % sum_add_split_nat_ivl
thf(fact_104_root__nonzero, axiom,
    ((![N : nat]: (~ (((fFT_Mirabelle_root @ N) = zero_zero_complex)))))). % root_nonzero
thf(fact_105_power2__less__imp__less, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ (power_power_nat @ X @ (numeral_numeral_nat @ (bit0 @ one))) @ (power_power_nat @ Y @ (numeral_numeral_nat @ (bit0 @ one)))) => ((ord_less_eq_nat @ zero_zero_nat @ Y) => (ord_less_nat @ X @ Y)))))). % power2_less_imp_less
thf(fact_106_Suc__n__div__2__gt__zero, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ zero_zero_nat @ (divide_divide_nat @ (suc @ N) @ (numeral_numeral_nat @ (bit0 @ one)))))))). % Suc_n_div_2_gt_zero
thf(fact_107_div__2__gt__zero, axiom,
    ((![N : nat]: ((ord_less_nat @ (suc @ zero_zero_nat) @ N) => (ord_less_nat @ zero_zero_nat @ (divide_divide_nat @ N @ (numeral_numeral_nat @ (bit0 @ one)))))))). % div_2_gt_zero
thf(fact_108_power__le__imp__le__base, axiom,
    ((![A : nat, N : nat, B : nat]: ((ord_less_eq_nat @ (power_power_nat @ A @ (suc @ N)) @ (power_power_nat @ B @ (suc @ N))) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (ord_less_eq_nat @ A @ B)))))). % power_le_imp_le_base
thf(fact_109_power__inject__base, axiom,
    ((![A : nat, N : nat, B : nat]: (((power_power_nat @ A @ (suc @ N)) = (power_power_nat @ B @ (suc @ N))) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (A = B))))))). % power_inject_base
thf(fact_110_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_complex @ zero_zero_complex @ N) = zero_zero_complex))))). % zero_power
thf(fact_111_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat))))). % zero_power
thf(fact_112_Suc__nat__number__of__add, axiom,
    ((![V : num, N : nat]: ((suc @ (plus_plus_nat @ (numeral_numeral_nat @ V) @ N)) = (plus_plus_nat @ (numeral_numeral_nat @ (plus_plus_num @ V @ one)) @ N))))). % Suc_nat_number_of_add
thf(fact_113_root__summation, axiom,
    ((![K2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ K2) => ((ord_less_nat @ K2 @ N) => ((groups59700922omplex @ (power_power_complex @ (power_power_complex @ (fFT_Mirabelle_root @ N) @ K2)) @ (set_or562006527an_nat @ zero_zero_nat @ N)) = zero_zero_complex)))))). % root_summation
thf(fact_114_power2__le__imp__le, axiom,
    ((![X : nat, Y : nat]: ((ord_less_eq_nat @ (power_power_nat @ X @ (numeral_numeral_nat @ (bit0 @ one))) @ (power_power_nat @ Y @ (numeral_numeral_nat @ (bit0 @ one)))) => ((ord_less_eq_nat @ zero_zero_nat @ Y) => (ord_less_eq_nat @ X @ Y)))))). % power2_le_imp_le
thf(fact_115_power2__eq__imp__eq, axiom,
    ((![X : nat, Y : nat]: (((power_power_nat @ X @ (numeral_numeral_nat @ (bit0 @ one))) = (power_power_nat @ Y @ (numeral_numeral_nat @ (bit0 @ one)))) => ((ord_less_eq_nat @ zero_zero_nat @ X) => ((ord_less_eq_nat @ zero_zero_nat @ Y) => (X = Y))))))). % power2_eq_imp_eq
thf(fact_116_less__2__cases__iff, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ (numeral_numeral_nat @ (bit0 @ one))) = (((N = zero_zero_nat)) | ((N = (suc @ zero_zero_nat)))))))). % less_2_cases_iff
thf(fact_117_less__2__cases, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ (numeral_numeral_nat @ (bit0 @ one))) => ((N = zero_zero_nat) | (N = (suc @ zero_zero_nat))))))). % less_2_cases
thf(fact_118_root__cancel, axiom,
    ((![D : nat, N : nat, K2 : nat]: ((ord_less_nat @ zero_zero_nat @ D) => ((power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ D @ N)) @ (times_times_nat @ D @ K2)) = (power_power_complex @ (fFT_Mirabelle_root @ N) @ K2)))))). % root_cancel

% Conjectures (1)
thf(conj_0, conjecture,
    (((groups59700922omplex @ (^[J2 : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ (plus_plus_nat @ i @ (times_times_nat @ i @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2)))) @ (a @ (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2))))) @ (set_or562006527an_nat @ zero_zero_nat @ m)) = (uminus1204672759omplex @ (divide1210191872omplex @ (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ i) @ (groups59700922omplex @ (^[J2 : nat]: (times_times_complex @ (power_power_complex @ (power_power_complex @ (fFT_Mirabelle_root @ m) @ i) @ J2) @ (a @ (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2))))) @ (set_or562006527an_nat @ zero_zero_nat @ m))) @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ m)))))).
